Equilibrium Risk Shifting and Interest Rate in an Opaque Financial System

Equilibrium Risk Shifting and Interest Rate in an

Opaque Financial System

Edouard Challe, Benoit Monjon, Xavier Ragot

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EQUILIBRIUM RISK SHIFTING AND INTEREST RATE IN AN

OPAQUE FINANCIAL SYSTEM

Edouard CHALLE

Benoit MOJON

Xavier RAGOT

Cahier n°

2012-19

ECOLE POLYTECHNIQUE

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE

DEPARTEMENT D'ECONOMIE

Equilibrium Risk Shifting and Interest Rate in an

Opaque Financial System

Edouard Challe

Benoit Mojon

Xavier Ragot

September 2, 2012

Abstract

We analyse the risk-taking behaviour of heterogenous intermediaries that are pro-tected by limited liability and choose both their amount of leverage and the risk expo-sure of their portfolio. Due to the opacity of the …nancial sector, outside providers of funds cannot distinguish “prudent” intermediaries from those “imprudent” ones that voluntarily hold high-risk portfolios and expose themselves to the risk of bankrupcy. We show how the number of imprudent intermediaries is determined in equilibrium jointly with the interest rate, and how both ultimately depend on the cross-sectional distribution of intermediaries’capital. One implication of our analysis is that an exoge-nous increase in the supply of funds to the intermediary sector (following, e.g., capital in‡ows) lowers interest rates and raises the number of imprudent intermediaries (the

We are particularly grateful to Andrew Atkeson, Regis Breton, Hans Gersbach, Michel Guillard, Robert Kollmann, François Larmande, Etienne Lehmann, Jose-Luis Peydro, Jean-Charles Rochet and Ken West for their comments. We also received helpful feedback from participants to the 2010 EEA meetings (Glasgow),the 2011 conference ‘Macroeconomic and Financial Intermadiation; Directions since the Crisis’ (Brussels) the 2011 Financial Intermediation Worshop (Nantes) and the third SciencesPo/Winsonsin School of Business Summer Macro-Finance Workshop. We also thank seminar participants at CREST, EM Lyon Business School, Banque de France, Bank of England and Ecole Polytechnique. Edouard Challe and Benoit Mojon acknowledge the …nancial support of the chair FDIR.

y_{Ecole Polytechnique, CREST and Banque de France; Email: edouard.challe@polytechnique.edu; Tel:} +33 (0)1 69 33 30 11; Fax: +33 (0)1 69 33 34 27 (corresponding author).

z_{Banque de France; Email: benoit.mojon@banque-france.fr.}

risk-taking channel of low interest rates). Another one is that easy …nancing may lead an increasing number of intermediaries to gamble for resurection following a bad shock to the sector’s capital, again raising economywide systemic risk (the

gambling-for-resurection channel of falling equity).

JEL codes: E44; G01; G20. Keywords: Risk shifting; Portfolio correlation; Systemic

risk; Financial opacity.

1

Introduction

The 2007-2010 …nancial crisis has rejuvenated the interest in systemic risk in the …nancial system, its dramatic spill over to the real economy and whether and how it should be addressed by public policies. We contribute to this debate with an analysis of the risk taking behaviour of …nancial intermediaries that have limited liabilities and may deliberately choose a level of risk in excess of the social optimum. We show how the level of economywide risk taking depends on the distribution of equity among intermediaries and the level of interest rate in the economy.

Our key assumption is that outside providers of funds cannot tell apart “prudent” and well diversi…ed banks from “imprudent” ones overly exposed to one particular asset, be-cause the balance sheets of individual intermediaries is imperfectly observable, or opaque. This assumption is consistent with the view of several commentators of the crisis including Brunnermeier (2009), Acharya and Richardson (2009), and Dubecq et al. (2009)1. In the decade prior to the crisis, risk transfer instruments, which have reached a very large scale in the U.S., have increased the opacity of banks’leverage and risk-taking incentives (Acharya and Schnabl, 2009). First, regulatory loopholes allowed banks evade capital requirements by securitising assets and providing (unregulated) liquidity support to “shadow” (i.e., o¤-balance-sheet) entities (Acharya and Richardson, 2009). Second, the …nancial sector as a whole e¤ectively repurchased much of the senior tranches of structured products, whose payo¤ distributions was particularly di¢ cult to assess (see, e.g., Coval et al., 2009). Third, some banks actively relied on “window dressing” to manipulate leverage …gures –by selling asset before the books releases to repurchase them at a later date (see, e.g., The Financial Crisis Inquiry Commission, 2011). Last but not least, this opacity may have taken the form

of shadow subsidiaries that were used to absorb poorly performing assets, as was revealed by the investigation on Lehman’s bankruptcy.2

While the opacity of the …nancial sector may have reached unprecedented level during the run-up to the crisis and the crisis itself, it has long been recognised as a key issue in that industry and one of the fundamental reasons for why it should be regulated. For example, Morgan (2002) shows that bond raters disagree signi…cantly more about U.S. …nancial in-termediaries than they do over other …rms, and interprets this result as evidence that banks are intrinsically more opaque –essentially because their assets are di¢ cult to observe and change at a fast pace.3 _{This feature of the industry severely limits the ability of outsiders}

(investors and rating agencies alike) to assess changes in bank’s capital structure in real time. To illustrate this point, Figure 1 reports the standard errors of an AR(2) regression of the idiosyncratic component of the capital-asset ratio of 90 French banks over the period 1993Q2-2009Q1. These standard deviations are sorted from the smallest to the largest. It is striking that for more than a quarter of these banks the one-quarter-ahead standard de-viation of the forecast error in the capital-asset ratio is higher than 2%. To summarise, the intrinsic nature of the banking industry combined with the recent trends in …nancial inno-vations have made it especially di¢ cult for outside providers of funds to accurately assess the true net worth level of individual banks.

When intermediaries’balance sheet is opaque, those with relatively low levels of capital may be tempted to hold high-risk asset portfolios, or even to gamble for resurrection in the 2_{On April 13, 2010, the New York Time reported that “Lehman Brothers operated a side business that}

allowed the defunct brokerage to transfer risky investments o¤ its books in the years leading up to its collapse, according to a report published Tuesday 13 April 2010. The …rm, called Hudson Castle, appeared to be an independent company, but played an important "behind-the-scenes role" at Lehman, .... Hudson is part of a "vast …nancial system" that operates largely beyond the reach of banking regulators. But banks can use such entities to raise cash by trading investments and, at times, make their …nances look arti…cially strong. The report said Lehman conducted several transactions greater than $1 billion with Hudson vehicles, but added that it is unclear how much money was involved since 2001. Critics charge that this type of creative …nancing allowed Lehman and other major banks to temporarily transfer risky investments in subprime mortgages and commercial real estate, the report said. While most of the deals done through operations such as Hudson are legal, the report points out that bank examiners have recently raised questions about other dubious accounting practices at Lehman.”

3_{See also Iannotta (2006) for similar evidence about European banks, and Flannery et al. (2010) on the}

Figure 1: Ranked forecast errors of an AR(2) of the idiosyncratic component of the capi-tal/asset ratio for 90 French banks. These are obtained in two stages. First, the capital-asset ratio of each of the 90 banks is regressed on their …rst three time-series common static factors (principal components). Second, the residual of this regression (which approximate the idio-syncratic movements of this ratio), is modelled as an AR(2) process, from which the standard deviation of the forecast error is computed. Data source: Banque de France, Commission Bancaire (see Jimborean and Mésonnier, 2010, for further desciption of the data).

face of worsening economic conditions. In our model, intermediaries’limited liability creates an incentives to correlate asset portfolios and raise leverage, thereby allowing intermediaries to raise their return on equity in case of success while transferring much of their losses to their creditors in case of failure. This tendency, however, is alleviated by intermediaries’ shareholders’initial equity stake, which disciplines risk-taking, thereby limiting leverage and favouring diversi…cation. We show that this trade-o¤ gives rise to an endogenous sorting of intermediaries along the equity dimension, with well capitalised intermediaries holding diversi…ed portfolios and keeping a limited level of leverage (that is, behaving prudently), and poorly capitalised ones heavily resorting to leverage to invest in correlated assets (i.e.,

behaving imprudently). Opacity implies that the former are not readily distinguishable from the latter, so that risk-prone behaviour may prosper without being immediately sanctioned by higher borrowing rates.

One property of our model is that the proportion of imprudent intermediaries and, there-fore, the level of systemic risk in the …nancial system, crucially depend on both the cross-sectional distribution of capital and the prevailing interest rate. The endogenous determina-tion of the number of imprudent intermediaries jointly with the (equilibrium) interest rate is our key contribution. Equipped with this joint equilibrium outcome, we analyse the impact on the interest rate and the number of imprudent intermediaries of two exogenous shocks: a lending boom that shifts the loan supply curve rightwards; and anequity squeeze that shifts the distribution of banks’ capital leftwards. As we show, the downward pressure on the equilibrium interest rate that follows the lending boom raises the number of imprudent in-termediaries and hence the level of economywide risk shifting (therisk-taking channel of low interest rates). An “equity squeeze”, that is, a reduction in the equity value of intermediaries’ shareholders after a negative aggregate shock, has the same e¤ect provided that the supply of funds is su¢ ciently elastic (the gambling-for-resurrection channel of falling equity). The evidence strongly suggests that both shocks occurred in the run-up to the current crisis. In the …rst half of the 2000 decade, both capital in‡ows from China and oil-exporting countries into the U.S. and the accommodative monetary policy of the Fed contributed to keep the yield curve very low; according to our model, this would have favoured imprudent behaviour by an increasing number of banks –those at the lower end of the capital distribution–thereby raising their risk exposure and the amount of systemic risk in the economy.4 Second, the tightening of U.S. monetary policy in 2004 and the rise in delinquency rates on subprime mortgages from 2006 onwards may have deteriorated the equity position of exposed interme-diaries, and hence favored gambling-for-resurection strategies. Landier et al. (2010) provide direct evidence of this behaviour for New Century Financial Corporation, a major subprime originator prior to its bankruptcy in 2007.5

4_{See Jimenez et al. (2010) for direct evidence that falling short-term interest rates favour risk-taking by}

banks that are at the lower end of the capital distribution.

5_{According to Landier et al. (2010), rising interest rate were conducive to more risk shifting, while many}

authors (e.g. Diamond and Rajan, 2009) suggests that low interest rates favor leverage and excess risk-taking. However, two e¤ects of interest rates on intermediaries’risk taking must be distinguished. First, holding the equity stake of bankowners …xed, low interest rates may indeed favor high leverage and risk taking. However,

Related literature. In our model, systemic risk in the …nancial sector arises from the interaction between i. intermediaries’limited liability and option to default (therisk shifting problem); ii. their incentive to correlate their risk exposure (the endogenous correlation problem); and iii. the di¢ culty for outside lenders to discriminate individual institutions on the basis of their true net worth level (the opacity problem). While our model is the …rst to explicitly connect these three dimensions, we build on many contributions that have studied each of them separately.

Our modelling of the risk shifting problem closely follows Allen and Gale (2000) and Acharya (2009), who show that limited liability leads …nancial institutions to overweight risky assets in their portfolio, relative to the …rst best.6 There are two main di¤erences between earlier models of risk shifting and ours. In Allen and Gale, market segmentation and limited-liability debt contracts twist intermediaries’risk-taking incentives and leads to an overvaluation (or “bubble”) in the price of the risky asset (a feature that is also in Challe and Ragot and Dubecq et al.). In Acharya, risky assets are in ‡exible supply so that their quantity (rather than price) adjust to clear the market. All these models share the property that intermediaries’ excessive risk-taking is ubiquitous: risky assets always have excessive space in intermediaries portfolios, leading all of them to be exposed to bankruptcy risk. We see this property as somewhat extreme, which leads us to emphasise the disciplining role of initial shareholders’ equity stake and to endogenise each intermediary’s (discrete) choice of adopting or not a bankruptcy-prone behaviour based on the expected costs and bene…ts of doing so. The second di¤erence with earlier contributions concerns the way we model excessive risk taking: while earlier models rely on intermediaries’overexposure to a risky asset relative to a safe one, in ours excessive risk taking exclusively takes the form of insu¢ cient portfolio diversi…cation in equilibrium.

This asset correlation problem has been the focus on several recent contributions, both empirical and theoretical. Acharya and Richardson (2009) notably document the overex-posure of the U.S. banking sector to securitised mortgages prior to the current crisis, with the risk associated with those securities being e¤ectively kept within the sector (via the use of unregulated liquidity enhancements or the repurchase of CDO tranches) rather than

rising interest rates lower asset values, which in turn deplete bankowners’net equity positions ex post and may trigger the gambling-for-resurection logic. Our model is consistent both views.

transferred to other investors and disseminated throughout the economy. Greenlaw et al. (2008) had reached similar conclusions. The dominant explanation for this excessive corre-lation, apparently at odds with standard …nance theory, is that it is natural consequence of the time-inconsistency of ex post bail-out or interest rate policies; namely, it is optimal to save banks ex post when a large number of them fails, which precisely occurs when they have chosen correlated portfolios in the …rst place –see Acharya and Yorulmazer (2007) and Fahri and Tirole (2010). Our model di¤ers from those in the source of moral hazard that leads to excess portfolio correlation, i.e., limited liability rather than time-inconsistent poli-cies. In Acharya (2009), the economywide correlation of risks arises from systemic failure externalities amongst intermediaries. The main di¤erence between Acharya’s endogenous correlation mechanism and ours is that in his framework banks are assumed to hold undiver-si…ed portfolio (because they are industry-speci…c lenders), and the puzzle to be explained is why correlation occursacross banks (i.e., why they tend to lend to the same industries). By contrast, in our model banks are unspecialised and choose the correlation of their portfolio at the individual level; but since those who opt for highly correlated portfolios favour the stochastically dominated asset, the very same asset is overinvested in at the aggregate level, hence more risk-taking at the individual level directly translates into greater systemic risk.

Finally, a number of authors have discussed the adverse consequences of the opacity of the …nancial sector for …nancial stability. The di¢ culty for (unsophisticated) outside lenders of perfectly observing bank assets is a traditional argument for why banks need to be supervised (e.g., Dewatripont and Tirole, 1994). More recently, Biais et al. (2010) have argued that …nancial innovations create asymmetric information problems that worsen the opacity of the …nancial sector. Our model focuses on one speci…c implication of opacity: the fact that outside providers of funds may …nd it di¢ cult to accurately measure bank shareholders’true stake and hence to adequately assess their risk-taking incentives.

The remainder of the paper is organised as follows. Section 2 introduces the model and characterises the optimal behaviour of intermediaries. Section 3 derives the equilibrium level of interest rate and systemic risk in the opaque economy, and carries out some comparative statics experiments. Section 4 uses a parameterised version of the model to analyse how noisy public signals about intermediaries’balance-sheet quality may e¤ect the equilibrium. Section 5 explores the consequences of imposing ‘naive’ capital ratios in this framework. Section 6 concludes.

2

The model

2.1

Timing, states and assets

There are two dates,t=_f1;2_g;two possible states at date 2,s=_fs1; s2g;and two

(supply-elastic) real assets available for purchase at date 1, a= _fa1; a2g. At date 1, loan contracts

are signed and investments in the real assets take place; at date 2, the state is revealed, asset payo¤s are collected and …nancial contracts are resolved –possibly via one party’s default. Any unit of investment in a1 pays R1 =Rh1 >0if s =s1 and 0otherwise, while any unit of

investment in a2 pays R2 = Rh₂ > 0 if s = s2 and Rl₂ > 0 otherwise.7 State s1 (s2) occurs with probability p(s1) p = 0:5 , > 0 (p(s2) = 1 p.) Finally the two assets are

assumed to have identical expected payo¤s, i.e.,

pRh₁ =pRl₂+ (1 p)R₂h: (1)

Our assumed joint payo¤ distribution has the following properties: when considered in isolation, a1 is more risky (in the sense of mean-preserving spread) than a2; however, the

strict negative correlation between the two assets implies that one of them may be used as a hedge against the portfolio risk generated by the other. In particular, a suitably diversi…ed portfolio pays the certain gross returnpRh

1 –thereby entirely eliminating bankruptcy risk for

a leveraged investor. This simple payo¤ structure allows us to focus on the joint choice of leverage and portfolio correlation as the ultimate source of endogenous aggregate risk in the economy.

2.2

Agents and market structure

There are two types of agents in the economy: “lenders” and “intermediaries”, both risk-neutral and in mass one. Our market structure (and implied decisions) is similar to that in Allen and Gale (2000) and Acharya (2009). In particular, markets are segmented, in the sense that intermediaries have exclusive access to the menu of assets a(due, for example, to

7_{Assuming that asset a}

1 has no liquidation value in state s2 greatly simplify the analysis of Section 4, where we need to sum up the repayments of heterogenous intermediaries to the lenders across states. Alternatively, one may assume that the asset has some liquidation value but that the latter cannot be recovered in case of default (due, e.g., to bankrupcy costs).

asymmetric information, di¤erence in asset management abilities, regulation etc.). Interme-diaries may borrow from the lenders to achieve their desired level of asset investment, and are protected by limited liability debt contracts. Once lending has taken place, the portfolio chosen by the intermediaries is out of the control of the lenders.

We modify this basic framework in two directions. First, we assume that an intermedi-ary’s funding partly comes out of inside equity, which will serve both to bu¤er the interme-diary’s balance-sheets against adverse shocks and to discipline its shareholders’risk-taking attitude.8 _{Second, we study the equilibrium of an economy populated by a large number}

of intermediaries with heterogenous equity levels that are imperfectly observed by outside providers of funds.

Intermediaries. Intermediaries’ shareholders maximise value, given their (exogenously given) initial equity stake e >0. Denoting by(xi)_i=1;2 0 the portfolio of an intermediary,

its balance sheet constraint may be written as:

P

xi e+b; (2)

with i= 1;2and whereb is the intermediary’s debt. Intermediaries face a convex, nonpecu-niary investment costc(Pxi), which satis…esc0(:)>0,c00(:)<0andc(0) = 0. For the sake

of tractability, our analysis in the body of the paper is carried out under the assumption thatc(:)is quadratic, but we show in Appendix B that all our results carry over to the more general isoelastic case. More speci…cally, c(:) takes the form:

c(Pxi) = (2 ) 1

(Pxi) 2

; >0: (3)

Given its initial equity stake eand a contracted gross interest rate r on borrowed funds, an intermediary chooses (xi) and, by implication, b –i.e., it chooses both the size and the

structure of the balance sheet. Limited liability implies that an intermediary’s payo¤ net of debt repayment is bounded below by zero, so the ex post net payo¤ generated by the portfolio(xi) is:

max [PxiRi rb;0]

8_{This paper focuses on agency problems between the intermediary’s owner-manager and its creditors,}

and hence abstracts from incorporating inobservability and con‡ict of interest between the owners and the managers. See Acharya et al. (2010) for a model of risk-shifting that explicitely incorporates both dimensions.

Substituting (2) (with equality) into the latter expression, we …nd the date 1 value (including the nonpecuniary cost) of an intermediary with initial equity e to be:

V (e) = max xi 0 P sp(s) (max [re+ P xi(Ri r);0]) c( P xi); (4) with s= 1;2.

In solving (4), intermediaries di¤er in the amount of the inside equity stake of its share-holders, e. The cross-sectional equity distribution is assumed to be characterised by a con-tinuous density function f(e; ) with support [0; emax] and c.d.f. F (e; ) =

Re

0 f(i; )di.

Since the number of intermediaries is normalised to one we have F(emax; ) = 1, while E Remax

0 ef(e; )diis the total capital of the intermediary sector. The parameter indexes

the location of the density function, with an increase in being associated with a rightward shifts in the distribution of equity level (so thatF (e; )<0.)

Lenders. Funds are supplied by households (the “lenders”), who lend their funds to the intermediary sector at date 1 in order to collect repayments at date 2. Each lender enjoys labour income w >0 at date 1 and maximisesu(c1) +c2s, where c1 is date 1 consumption, c2s consumption at date 2 in state s, and u(:) a twice continuously di¤erentiable, strictly

increasing and strictly concave function. Let s denote lenders’ ex post date 2 return in

state s from lending to the intermediary sector, and P_sp(s) s the corresponding

ex ante return (Note that both in general di¤er from the face lending rate r due to the possibility of intermediaries’ default.) Lenders choose their loan supply Bs_{, where B}s ₌

arg maxu(c1) +

P

sp(s)c2s, subject to c1 =w B and c2s =B s. The implied loan supply

curve is:

Bs( ;w) = w u0 1( ); (5)

which is continuous and strictly increasing in both arguments. In short, risk neutrality implies that lenders value the expected return on loans, , with the implied loan return curve being shifted by date 1 income,w. We impose speci…c parameter restrictions later on ensuring thatBs( ;w)>0 in equilibrium.

2.3

First best

The key contractual friction in this economy is that an intermediary maximises the expected terminal payo¤ to its (risk-neutral) shareholders who are protected by limited liability –

and hence transfer losses to the debtors in case of default. Before further analysing the implications of this distortion, we compute the …rst-best outcome, where this distortion is removed.

Planner’s problem. Since a1 and a2 have identical expected payo¤ and are perfectly

negatively correlated, a fully diversi…ed portfolio that entirely eliminates the payo¤ risk is always e¢ cient, at least weakly. A portfolio(xi)_i=1;2 pays the certain payo¤(Pxi)pRh1 if it

pays identical payo¤s across states, that is, if it satis…es

x1Rh1 +x2Rl2 =x2Rh2; (6)

where the left and right hand sides are the portfolio payo¤s in statess1 and s2, respectively.

Equation (6) implies that the riskless portfolio must be composed of the following asset shares: ^ x1 P ^ xi =p R h 2 Rl2 Rh 2 ; _Px^2 ^ xi =p R h 1 Rh 2 : (7)

Equation (7) characterises the structure of the portfolio but not its optimal size. Let ~

C denote the total consumption of intermediaries (’shareholders), andC1 and C2 the total

consumption of lenders at date 1 and 2, respectively. The planner solvesmaxu(C1)+C2+ ~C, subject to the following …rst- and second-period resource constraints:

(Px^i) +C1 =w+E; C2+ ~C = (Px^i)pRh1 c(

P

^

xi):

Hence, the optimal level of investment Px^i satis…es:

P ^ xi = pRh1 u0(w+E P ^ xi) : (8)

Decentralisation. Since the limited-liability constraint is the only friction a¤ecting in-termediaries portfolio choice, the planner’s problem can be decentralised by removing this constraint –or equivalently, by punishing default su¢ ciently severely. When the option to default is not operative, the value of an intermediary in (4) becomes:

^ V (e) = max xi 0 re+ (Pxi) pRh1 r c( P xi); (9)

where we have used the fact that P_sp(s)Pxi(Ri r) = (

P

xi) pRh1 r under full

P

^

xi = pRh1 r . Combined with (1) and (6), we …nd it to be:

(^x1;x^2) = pRh1 r p Rh 2 Rl2 Rh 2 ; pRh₁ r pR h 1 Rh 2 : (10)

Portfolio (10) is only weakly dominant because, due to agents’risk neutrality, a continuum of portfolios in fact achieve the same welfare outcome as (10). Indeed, for lenders to enjoy the face returnrin both states, it is enough that intermediaries besolvent in both states, which is possible (to some extent) with an imperfectly diversi…ed portfolio thanks to the bu¤ering role of intermediaries’capital. The solvency conditions impose that a given portfolio(xi)i=1;2

never generates a negative net payo¤ ex post, i.e.,

re+Pxi(Ri r) 0; s=s1; s2; (11)

where (R1; R2) = Rh1; Rl2 ifs =s1 and 0; Rh2 ifs =s2. Combining (11) with the optimal

balance-sheet size Px^i = pRh1 r , we …nd that a solvent portfolio must be such that x1 2[x1; x1];0< x1 < x1 <1, where x1 and x1 are:

x₁ = pR h 1 r r Rl2 re Rh 1 Rl2 ; x₁ = pR h 1 r Rh2 r +re Rh 2 ; and where x₁ < pRh

1 r whenever the intermediary is leveraged. In short, given the

optimal balance-sheet size pRh _{r ; x}

1 cannot be too low, otherwise the intermediary

would default in state s1; it cannot be too high either (and hence x2 too low), otherwise

default would occur in state s2. Intermediaries that deviate from the riskless portfolio (^xi)

while still satisfying Px^i = pRh1 r and x1 2 [x1; x1] will bear some asset risk but are

indi¤erent to it thanks to risk neutrality. These portfolios, which we refer to as “prudent”, lie along a closed subinterval of thex2 = pR₁h r x1 line and include the riskless portfolio (^xi) –see Figure 1(a) below.

Equilibrium. To complete the characterisation of the …rst-best outcome, we must compute the equilibrium interest rate r^that results from the equality of the aggregate demand and supply for loanable funds. Since Px^i = pR1h r , it follows that the leverage of an

inter-mediary with inside equityeand facing the interest rateris given by^b(r; e) = pRh₁ r e. The aggregate demand for funds is obtained by summing up the demands for debt by all intermediaries, i.e., ^ Bd(r; ) = Z emax 0 ^ b(r; e)dF (e; ) = pRh₁ r E: (12)

On the other hand, since intermediaries never default in the …rst-best equilibrium, lenders are repaid r with certainty. Hence, we may rewrite (5) as:

^

Bs(r;w) = w u0 1(r): (13)

^

Bd_{(r; ) is continuous and linearly decreasing in r, while B^}s_{(r;w) is continuous and}

strictly increasing in r (since u00(:) < 0). Hence the two curves cross at most once and, if they do, give a unique equilibrium interest rater^. In the remainder of the paper, we focus on equilibria in which all intermediaries are active and leveraged. Lemma 1 provides a su¢ cient condition for the existence of a …rst-best equilibrium with this property.

Lemma 1. Assume that i. pRh

1 > emax and ii. w > emax E+u0 1 pRh1 emax= :Then,

the …rst-best equilibrium is unique and such that^b(r; e)>0for all e₂[0; emax].

All proofs are in the Appendix. Essentially, a unique equilibrium with all intermediaries being leveraged exists if both expected asset payo¤s (i.e.,pRh

1) and lenders’income (i.e.,w)

are su¢ ciently large. This equilibrium is depicted in Figure 2(b) below.

3

Loanable funds equilibrium under risk-shifting

3.1

Intermediaries’behaviour

The presence of the limited-liability debt contracts a¤ect investment choices by altering intermediaries’ shareholders payo¤s relative to the …rst best. Namely, value maximisation under limited liability may lead an intermediary to choose a high risk/high expected payo¤ strategy, thereby maximising its own payo¤ in case of success while transferring losses to the lenders in case default.

We work the problem of an intermediary (i.e., equation (4)) backwards. Let us refer to as “prudent” an intermediary whose asset portfolio satis…es both solvency constraints in (11), and denote its value as V (e). Similarly, let us call “imprudent” an intermediary whose portfolio violates one of the two inequalities in (11) –thereby triggering default in one of the two states–, and denote its value byV (e). The intermediary chooses the best option, giving a value to the initial equity holders ofV (e) = max [V (e); V (e)].

Prudent intermediaries. Trivially, the absence of the option to default implies that the portfolio choice of a prudent intermediary is the same as in the …rst best:

P

x_i = pRh₁ r ; x₁ < x₁ < x₁; (14)

b (r; e) = pRh₁ r e: (15)

Substituting (14)–(15) into (9), we …nd the value of a prudent intermediary to be:

V (e) = re+ ( =2) pR₁h r 2 (= ^V (e)): (16)

Imprudent intermediaries. Imprudent intermediaries, unlike prudent ones, correlate their portfolio and consequently default in one of the two states. Consider …rst the optimal portfolio choice of an intermediary having chosen to overweight asset a1 in its portfolio, and thus defaults at date 2 if state s2 occurs. Ex ante, this intermediary earns zero with

probability 1 p, so the objective (4) becomes: max

xi 0

V (e) =p re+x1 R1h r +x2 Rl2 r c(

P

xi) (17)

Sincex1 andx2 enter symmetrically in the cost function whileRh₁ > Rl₂, the intermediary must entirely disregard a2, leading to the optimal portfolio:

(x₁ ; x₂ ) = p Rh₁ r ;0 ; (18)

b (r; e) = p Rh₁ r e: (19)

An alternative investment strategy for an imprudent intermediary would be to overweight

a2;and hence to default ifs1 occurs ex post. However, it is straightforward to show that it is

never optimal to do so under our distributional assumptions. Indeed, imprudent behaviour implies that the intermediary earns zero if the wrong state occurs, and accordingly only val-ues the state corresponding to the asset being invested in. Since the univariate distribution of a1 is a mean-preserving spread of that of a2, a1 has more value to the imprudent

inter-mediary than a2.9 Substituting (18) into (17), we …nd the optimised value of an imprudent

intermediary to be:

V (e) =pre+ ( =2) p Rh₁ r 2: (20)

9_{An intermediary choosing the default in state s}

1 does not value payo¤s in that state and hence max-imises (1 p) x1(0 r) +x2 Rh2 r +re c(x1+x2), leading to the optimal portfolio (~x1 ;x~2 ) = 0; (1 p) R2h r : Computing and comparing the ex ante utility levels associated with (x1 ; x2 ) and (~x₁ ;x~₂ )leads the former to be prefered, provided that is not too large.

To summarise, imprudent intermediaries have two distinguishing characteristics, relative to prudent intermediaries. First, they perfectly correlated their asset portfolio (since x₂ = 0), thereby maximising both their payo¤ in case of success and lenders’ losses in case of default. Second, they endogenously choose a larger balance sheet size (since x₁ > Px_i), which in turns implies more leverage for any given level of equitye(i.e., b (r; e)> b (r; e)). This latter property is a direct implication of the fact that imprudent intermediaries avoid repayment with probability 1 p: This e¤ectively lowers the cost of debt ex ante for any given face interest rater, relative to the cost faced by prudent intermediaries (who repay in both states). In the (x1; x2) plane, the imprudent portfolio lies on the x1 axis and the left

of thex2 = pRh₁ r x1 line –see Figure 1(a).

x1 x1* Prudent portfolios Riskless portfolio x2 x₁** Imprudent portfolio * 1 x V(e) V*(e) V**(e) e max[V*(e),V**(e)] ) ( ~ _r e (a) (b)

Figure 1: Intermediaries’optimal portfolios (a) and value (b).

Value of an intermediary. Expressions (16) and (20) re‡ect the joint roles of equity and the borrowing rate in a¤ecting the intermediary’s value and thus incentives to behave prudently or imprudently. For a given level of equity and borrowing rate, imprudent in-termediaries buy larger portfolios, consequently earn large payo¤s in case of success, which goes towards raising value (see the second term in the right hand side of both expressions);

however, they also risk losing their equity (with probability 1 p), which tends to reduce value for any given initial equity stake (the …rst term). Comparing (16) and (20) and assum-ing that indi¤erent intermediaries behave prudently, we …nd that an intermediary engages in imprudent behaviour whenever its equity state is su¢ ciently low, that is, if and only if:

e <~e(r) pRh₁ 1 +p

2 r (21)

Equation (21) implies that a poorly capitalised intermediary, i.e., one with low equity stake and hence relatively little to lose in case of default, will engage in imprudent behaviour, while an intermediary with high shareholders’equity stake, and hence much to lose in case of default, will behave prudently. The implied value of an intermediary as a function of e, i.e., V (e) = max [V (e); V (e)]is depicted in Figure 1(b).

A key implication of (21) is that lower borrowing rates raise the cut-o¤ equity level below which the intermediary chooses to behave imprudently. To further understand why this is the case, compare the impact of a marginal rise inr onV (e)and V (e)–that is, for each strategy, the loss in the intermediary’s value associated with a rise in the face …nancing cost. Using (16) and (20), we …nd these falls to be:

V_r (e) = b (r; e); V_r (e) = pb (r; e):

These expressions follow from the envelop theorem and have a straightforward interpre-tation. For the prudent intermediary, who never defaults and hence always repays r per unit of debt, the loss in value associated with a marginal rise in r is its total amount debt,

b (r; e):For the imprudent intermediary, who only repays in state 1, the loss in value is the relevant amount of debt, b (r; e), times the probability that it will actually be repaid, p. For a rise in r to lower the threshold e~, it must be the case thatV (e)increases more than

V (e) for the marginal intermediary, i.e., that for whom V (~e) = V (~e) (i.e., that inter-mediary must turn prudent, rather than imprudent, following a rise in the interest rate). It must be the case that V_r (~e)> V_r (~e)or, equivalently by using the two expressions above,

b (r;~e)< pb (r;e~): a switch by the marginal intermediary from the prudent to the impru-dent investment strategy must involve a su¢ ciently large increase in leverage. This property can be shown to hold not only in the quadratic case but also for any isoelastic investment cost function (see Appendix B for details).

3.2

Aggregate demand for funds

Our key assumption here is that while the distribution of equity levels is perfectly known by outside lenders, …nancial opacity prevents lenders from observing the equity levels any particular intermediary. Hence, lenders cannot condition the loan rate on the speci…c equity level of an intermediary, so that a single borrowing rate r applies to the entire market.10

Then, we may de…ne

g(r; )

Z ~e(r) 0

f(e; )de =F (~e(r) ; ) (22) as the proportion of imprudent intermediaries in the economy at a given interest rate r. Note thatgr(r; ) = (1 +p)f(~e(r) ; )=2<0;that is, a lower face interest rate raises the

proportion of imprudent intermediaries in the economy by increasing the threshold equity level e~(r). Moreover, we have g (r; ) = F (~e; ) < 0, that is, an increase in lowers the proportion of imprudent intermediary (for any given value of the cut-o¤e~(r)).

The total demand for funds aggregates the leverage choices of individual intermediaries, appropriately weighted by their shares in the economy. It is thus given by:

Bd(r; ) = Z e~(r) 0 b (r; e)dF (e; ) + Z emax ~ e(r) b (r; e)dF (e; ): (23) Equation (23) shows that the interest rate will a¤ect the demand for loanable funds in two ways: …rst, it will a¤ect the demand for funding of every single intermediaries (the ‘intensive’ leverage margin); and second, by shifting the threshold e~(r), it will cause a discontinuous change in the leverage choice of some of them, from prudent to imprudent or the other way around (the ‘extensive’ leverage margin.) Substituting (15) and (19) into the latter expression, using (22) and rearranging, the total demand for funds is found to be:

Bd(r; ) = pRh₁ r(1 (1 p)g(r; )) E: (24) In the(B; r)plane, theBd(r; )curve lies to the right of the B^d(r; )curve, its …rst-best counterpart. This is because, for any given value of r, the risk-shifting equilibrium includes a nonnegative fraction of imprudent intermediaries, whose demand for debt is larger than that of prudent intermediaries at any given interest rate r (see Figure 2(b)).

10_{We assume for simplicity that intermediaries are completely identical from the point of view of the}

lender. Our result carry over in a set-up with partially segmented market involving di¤erent groups of intermediaries, with the members of each group facing the same interest rate. What matters for our results is the presence of an unbserved residual heterogeneity in interminedaries’equity stake.

There are two properties of the aggregate demand for loanable funds that are worth discussing at this stage. First, it is continuous and decreasing in the borrowing rate, i.e.,

B_rd(r; ) = (1 (1 p)g(r; )) + (1 p)rgr(r; )<0:

Two factors contribute to make the demand for funds a downward-sloping function of r. First, a lower interest rate raises the leverage of both prudent and imprudent intermediaries –see the optimal investment rules (14) and (18). Second, a lower interest rate induces “marginal” intermediaries (those which are close to the cut-o¤ equity level e~ in (21)) to switch from prudent to imprudent behaviour, and those experience a discontinuous increase in their leverage –again, by (14) and (18). Hence, changes in the borrowing rate a¤ect the “intensive” (i.e., conditional on not switching behaviour) and “extensive” (i.e., the number of intermediaries who switch behaviour) leverage margins in the same direction.

The second relevant property of the curve is that, holdingrconstant,Bd_{increases as the}

distribution of equity shifts leftwards. That is,

Bd(r; ) = r(1 p)g (r; )<0:

This is because, as the equity level of intermediaries decreases, some of them switch from prudent to imprudent behaviour. As imprudent intermediaries choose higher leverage than prudent ones, this composition e¤ect translate into an upward shift in the aggregate demand for funds.

3.3

Aggregate supply of funds

The aggregate supply of funds depends on the expected return on loans, , which under risk shifting not only depends on the face borrowing rate but also on both the share of imprudent intermediaries and the probability that they go bankrupt. In state 1, which occurs with probability p, all intermediaries repay the face interest rate r to the lenders: prudent intermediaries because they are always able to, imprudent ones because their risky bets turned out to be successful. In state 2, which occurs with complementary probability, only prudent intermediaries, which are in number1 g(r; ), are able to repayr. Imprudent intermediaries’ bets, on the contrary, turn out to be unsuccessful, leaving lenders with no repayment at all. Summing up unit repayments across states and intermediaries types and

rearranging, we …nd the ex ante gross return on loans to be:

(r; ) =pr+ (1 p) (1 g(r; ))r (25) =r(1 (1 p)g(r; )):

Note that this ex ante return is strictly increasing in the face interest rate, i.e.,

r(r; ) = 1 (1 p)g(r; ) (1 p)rgr(r; )>0: (26)

The increasingness of (r; )with respect toroccurs for two reasons. First, a higher face interest rate increases intermediaries’repayment if they do not default (the1 (1 p)g(r; )>

0part of (26)). Second, a higher face interest rate favours prudent rather than risky behav-iour by raising the threshold e~, and hence by lowering the probability of default on a loan unit (the (1 p)gr(r; )>0part). It follows that for( ; w)given the loan supply function

is a nondecreasing, continuous function of r, which we may express as:

Bs(r; ; w) =w u0 1(r(1 (1 p)g(r; ))): (27) Let us brie‡y summarise the properties of the aggregate supply curve, before we analyse the equilibrium in the market for loanable funds. First, Bs_{(r; ; w) is strictly increasing in} r; holding ( ; w) constant; this follows from (5), the strictly concavity of u(:), and the strict monotonicity of w.r.t. r (see (26)). Second, from (5) it is strictly increasing inw, holding

r and constant. Third, it is increasing in , holding r and w constant. The reason for this is that a higher overall level of equity in the economy raises the number of prudent intermediaries (i.e., g (r; )<0), and hence the expected return on loans (see (25)).

In the (B; r) plane, theBs_{(r; ; w)curve lies to the left of its …rst-best analogue,B}^s_{(r; ; w):}

This is because in the equilibrium with risk shifting lenders expect a nonnegative fraction of intermediaries to go bankrupt if states2 occurs. Hence, any given value of the face interest

rate r is associated with a lower expected return in the risk-shifting equilibrium than in the …rst-best –and hence with a lower supply of loanable funds (see Figure 2(b)).

3.4

Market clearing

In equilibrium, the total demand for funds by the intermediary sector must equal the total supply of funds provided by outside lenders. In other words, the face interest rate that clears

the market for loanable funds must satisfy

Bs(r; ; w) = Bd(r; ) (28)

Since Bd_{(r; )is continuously decreasing inr whileB}s_{(r; ; w)is continuously increasing}

inr, the equilibrium is unique provided that it exists. Again, we are focusing on risk-shifting equilibria in which all intermediaries are leveraged, the conditions under which this is the case being summarised in the following lemma.

Lemma 2. Assume that i. pRh

1 > emax and ii.

w >max u0 1 pRh E= ; r(1 p)g((r; )) E+u0 1(r(1 (1 p)g(r; ))) ;

where r = pRh₁ emax= : Then, the equilibrium with risk shifting is unique and such that b (e)and b (e) are positive fall alle₂[0; emax].

To summarise, the equilibrium is well behaved provided that lenders’income, w; is suf-…ciently large. The existence conditions stated in Lemma 2 are slightly more stringent than those stated in Lemma 1, so the former also ensure the existence of the …rst-best outcome characterised in Section 2.3.

The equilibrium in the market for loanable funds is depicted in Figure 2(b). The inter-section of the two curves gives the equilibrium contracted loan rate r, given the exogenous parameter set ( ; w). The loan rate in turn determines the equilibrium share of imprudent intermediariesg(r; )(by equation (22)), as well as the equilibrium expected return on loans to intermediaries, (r; ) (by (25)). Note that despite di¤erences in the implied equilibrium interest rate in the two economies, the equilibrium amount of aggregate lending is the same. Indeed, the interest rate in the …rst-best equilibrium satis…es pRh

1 r^ E =w u0 1(^r);

while the expected rate of return in the risk-shifting equilibrium satis…es pRh₁ E =

w u0 1_{( ). This implies that = ^r (i.e., lenders’expected compensation for their loans in}

the same across the two equilibria), so that Bs_{( ;w) = ^B}s_{(^r;w) (i.e., they lend the same}

amount). From (25) and the fact that = ^r, we …nd the interest rate premium generated by the presence of imprudent intermediaries to be:

r

^

r =

g(r; )

(1 p) 1 g(r; );

which is positive and increasing in the both the number of such intermediaries, g(r; ), and the probability that they go bust, 1 p.

f

(

e

;

ε

)

e

(

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



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





−

+

=

2

1

~

r

p

pR

e

θ

g

(

r;

ε

)

B

(

r;

ε

)

B

(

r;

ε

, w

)

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r

(a)

(b)

)

,

;

(

ˆ

r

w

B

ε

)

;

(

ˆ

r

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4

Impact of aggregate shocks

We may now state the main predictions of the model about how shifts in the underlying fundamentals (the supply of funds and the distribution of intermediaries’capital) a¤ect the three key equilibrium variables, r, (r; )and g(r; ): These predictions are summarised in the following propositions.

Proposition 1 (Lending boom). An exogenous increase in the supply of funds (i.e.,

dw >0) i) lowers the equilibrium contracted rate,r, ii) lowers the expected return on loans, (r; ), and iii) raises the share of imprudent intermediaries in the economy, g(r; ).

Proposition 1 essentially states that easier …nancing conditions for intermediaries tend to fuel systemic risk by inducing an increasing number of intermediaries to take larger and riskier bets; conversely, tighter credit raises the interest rate and discipline banks’risk-taking behaviour. The e¤ect of the boom shift in the funds supply curve is depicted in Figure 3(a). More speci…cally, the boom is associated with a rightward shift in theBs _{locus, whose direct}

e¤ect is to lower the equilibrium contracted loan rate. Holding constant, the new value of r is associated with a lower value of the equity cuto¤e~(r) in (21); so that a increasing number of intermediaries turn from prudent to imprudent –i.e.,g(r; ) rises. Both the lower value of r and the higher value of g(r; ) contribute to lower the expected return on loans,

(r; ).

While our analysis remains formal, several interpretations may be given to the shift in credit supply leading to easier …nancing conditions. According to Bernanke (2005), for ex-ample, a supply-driven shift in funding occurred in the …rst half of the last decade due to recycled balance-of-payment surpluses from China and oil-exporting countries; in this inter-pretation, systemic risk in the U.S. was closely related to the “global imbalances” problem, which was itself rooted in the willingness of surplus countries to hoard wealth in the form of U.S. assets. Another view has it that exceptionally loose monetary policy leading to exceed-ingly low real interest rates in the wake of the 2001 recession in the U.S. would have given rise to a “risk-taking”channel of monetary policy, thereby fostering widespread systemic risk in the U.S. …nancial sector (see Taylor, 2009, Adrian and Shin, 2010, as well as Altunbas et al. (2010) for a survey and some evidence).11 Be it the consequence of either or both, the 11_{As argued by Obstfeld and Rogo¤ (2009), these two views are likely more complementary than}

model unambiguously predicts that falling interest rates raise risk-taking by an increasing number of banks and hence the economywide level of risk. Moreover, the model predicts that this increase in aggregate risk is rooted in changes in the portfolio choices of less capitalised intermediaries –i.e., those to the left of, but close to, the equity cuto¤e~(r). This channel is consistent with the …ndings of Jiminez et al. (2010), who study the risk-taking behaviour of a panel of Spanish banks and …nd that falling short-term rates increase risk-taking low-capital banks (rather than the “average” bank.)

Proposition 2 (Equity squeeze). A downward shift in the distribution of equity (i.e.,

d < 0) raises the equilibrium interest rate, r. If the elasticity of the credit supply with respect to is su¢ ciently high, then it also raises the share of imprudent intermediaries,

g(r; ).

Proposition 2 re‡ects the three e¤ects at work following a downward shifts in the dis-tribution of equity. First, for a given value of the cut-o¤~e, the shift directly increases the number of imprudent banks in the economy by lowering the stake of “marginal” interme-diaries (i.e., those who are initially to the right of, but close to, ~e); those intermediaries then discontinuously raise their leverage while engaging in imprudent behaviour (see (14) and (18)), thereby raising the demand for funds. Second, to the extent that this shift lowers the overall equity base of the intermediary sector, E, all intermediaries, which have a target portfolio size, seek to o¤set the loss in internal funding by external debt, again raising the economywide demand for funding. Both of these e¤ects shift the Bd_{-curve rightwards and}

exert an upward pressure on the equilibrium borrowing rate, r. Third, this increase in the borrowing rate has a disciplining e¤ect on the intermediary sector by shifting the cut-o¤ equity level e~leftwards. Hence, while the e¤ect of the equity squeeze on the borrowing rate is not ambiguous, that on the share of imprudent intermediaries is. However, if the supply of funds is su¢ ciently elastic, the adjustment of the borrowing rate after the shock and its disciplining e¤ect will be limited, causing g(r; ) to rise.

This situation is depicted in Figure 3(b). The initial distributional shift causes the

g(r; ) curve to shift leftwards. The direct impact of higher risk (holding r …xed) is to lower the expected return on loans, (r; ), which in the (B; r) plane manifests itself as an exogenous reduction in lending (i.e., an inwards shift of the Bs _{curve). Finally, the increase}

is su¢ ciently elastic (that is, the slope of the Bs curve is su¢ ciently low), then the overall e¤ect of the three shifts is to raise the equilibrium value of g(r; )

r

g

(

r

,

ε)

B

(

r;

ε

,w

)

B

(

r

;

ε

)

(a)

Lending boom

r

g

(

r

,

ε)

(b)

Equity squeeze

(elastic supply of funds)

B

(

r

;

ε

)

B

(

r;

ε

,w

)

B

B

Figure 3. Impact of a lending boom (a) and a equity squeeze (b) on the interest rate,r, and the share of imprudent intermediaries, g(r; ).

5

Information about intermediaries’balance-sheets and

endogenous market segmentation

Our analysis above emphasise intermediaries’ balance-sheet opacity as a major source of systemic risk. The key mechanism is that the unobservability of balance sheets makes it possible for bad banks to pretend to be good banks, implying that clearing of the market for loanable funds operates in a single market and at a single interest rate. In order to make this channel as transparent as possible, we derived our results under the somewhat extreme assumption that all intermediaries look alike from the point of view of outside lenders. In reality, some public (and private) information about intermediaries’balance-sheet is available that may mitigate the opacity problem. In this Section, we extend our analysis to allow for (noisy) signals about intermediaries’s capital, which naturally generates a di¤erentiation of the market for loanable funds –in as much a ‘good’ and ‘bad’ intermediaries can to some extent be recognised as such. For the sake of tractability we illustrate this possibility by means of a simple parametric example, but we conjecture that the properties that come out of this exercise hold much more generally.

Distributions, signal structure and parameters. We assume here that the uncon-ditional distribution of intermediaires’ inside equity is uniform with support [0;1], so that

f(e) = 1, F (e) = e and E = 1=2. Outside lenders receive the following symmetric bi-nary signal about every intermediary: if e 1=2, then = g (‘good’) with probability 2 [1=2;1) and = b (‘bad’) w.p. 1 . Symmetrically, if e < 1=2, then = b with probability and =g w.p. 1 . Under these assumptions, the marginal density of the signals is simply Pr ( =h) = Pr ( =l) = 1=2. From Bayes’ rule, the observation of the signal produces the following two conditional distributions and cumulative density functions (both of which are indexed by the signal quality ):

f(e_jg; ) = 8 < : 2 (1 ) fore < 1₂ 2 for e 1₂ ; F (e_jg; ) = 8 < : 2 (1 )e for e < 1₂ 1 2 + 2 e fore 1₂ (29) f(e_jb; ) = 8 < : 2 for e < 1 2 2 (1 ) fore 1₂ ; F (e_jb; ) = 8 < : 2 e fore < 1 2 2 1 + 2 (1 )e fore 1₂ (30)

Note that the quality of the signal encompasses two limit cases. When = 1=2, the signal is uninformative and the two conditional distributions coincide with the unconditional one. When _!1, in the limit the signals exactly identify every intermediary as belonging to the upper or the lower halves of the distribution. Regarding the other deep parameters, we set

p =w = 1=2; = 1, Rh

1 = 4: Finally, we focus on the case whereu(:) = 0, so that lenders

only value terminal consumption and inelastically lend w to the intermediary sector.

The signals identify two categories of intermediaries, and hence two separate markets for loanable funds, each with their own face interest rate. In each market, the problem of an individual intermediary in is similar to that described in Section 3, except that they now take their own face borrowing rater as given. Under the assumed parameters, an intermediary borrowing in market ; =b; g; behaves prudently if and only if

e e~(r ) = 2 3r

4 ; (31)

while the share of imprudent intermediaries in that market is given by:

g (r ; ) =F ( ~e(r )_j ; );

where F ( ~e(r )_j ; ) is determined by (29)–(31).

We may now compute the demand for funds in each market by integrating intermediaries leverage choices as in (24). Under our parameters, the demand for funds in market is

Bd; (r ; ) = Z e~(r ) 0 b (r ; e)dF (e_j ; ) + Z 1 ~ e(r ) b (r ; e)dF (e_js; ) = 2 r 1 1 2g (r ; ) E ; (32)

where, from equations (29)–(30),

Eb = Z 1 0 ef(e_jb; )de= 3 2 4 ; E g ₌ Z 1 0 ef(e_jg; )de = 1 + 2 4 :

Uninformative signals. Let us …rst solve for the equilibrium face interest rate and share of imprudent intermediaries in the uninformative case (i.e., = 1=2), which corresponds to the baseline model analysed in the previous Sections (since in this situation both conditional equity distributions coincide with the unconditional one, and we are back to the single-market case.) Under our parameters speci…cation, the (unique) face interest rater is determined by the following equilibrium condition:

2 r 1 1 2g r; 1 2 1 2 = 1 2; (33)

where the left hand side is the aggregate demand for loanable funds and the right hand side is the aggregate supply of loanable funds,w= 1=2. Using (25), we may then explicitly solve for the expected return on loans, which is given by:

r;1

2 = 1 :

By equation (25) again, the face interest rate must satisfy r(1 F(~e(r))=2) = 1. Since the unconditional CDF isF(e) = eande~(r)is given by (31), we get the equilibrium interest rate r=p8=3 = 1:633; which in turns produces a share of imprudent intermediaries of

g r;1 2 = 2

3

4r= 0:775 (34)

Informative signals. We …rst note that even when >1=2and markets are di¤erentiated, by no-arbitrage and given lenders’risk neutrality, the expected return on loans in the two markets must be identical, i.e., we must have g_(rg_{; ) =} b _rb_{; ~ r}g_{; r}b_{; . Second,}

by (25) this common expected rate of return satis…es ~ rg; rb; =r 1 1

2g (r ; ) ; =b; g: (35)

Using (32) and (35), we may express the demand for loanable funds in market as

Bd; (r ; ) = 2 ~ rg; rb; E : (36) The marginal density of the signal isPr ( =h) = Pr ( =l) = 1=2), so that total equity is

Eb_+Eg _{=2 = E while the total demand for loanable funds is (B}d; _{(r ; ) +B}d; _{(r ; ))=2.}

The latter must sum up to the aggregate supply of loanable fundsw= 1=2; so from (36) we get

~ rg; rb; = 1 =

In can be shown (by contradiction) that for all ₂ [1=2;1) and = b; g, we always have e~(r ) > 1=2, so the upper halves of the conditional cumulative distribution functions in (29)–(30) determine the shares of imprudent intermediaries in each market. This implies that these shares are given by:

gb rb; =F e r~ b b; _e~(r)>1=2 = 2 1 + 2 (1 ) 2 3r b 4 ; (37) gg(rg; ) =F ( ~e(rg)_jg; )_e~(r)>1=2 = 1 2 + 2 2 3r g 4 : (38)

Finally, in both markets we have (r ; ) = r (1 g rb; =2). Using (37)–(38), the fact that (r ; ) = 1; =g; b and rearranging, we …nd that rb_{; r}g _solve:

1 = rb 1 2+ 3 (1 )rs 4 ; 1 = r g 1 2 + 3 4 r s :

Then, with rb_{; r}g _{known, we may compute the shares of imprudent intermediaries in}

each market from (37)–(38), and that in the whole economy(gb rb; +gg(rg; ))=2:When = 1=2, we have rb _{= r}g _{= r =} p_{8=3 (the uninformative limit studied above.) As}

rises above = 1=2 –i.e., the signal becomes more and more informative–, rb _{goes up and} rg goes down –since low- versus high-equity intermediaries are more and more identi…ed as such. Figure 4 plots the two face interest rates as a function of ; as well as the shares of imprudent intermediaries in the two markets and the implied proportion of such intermedi-aries economywide. In this example, the more informative the signal, the higher the share of imprudent intermediaries in the economy. To understand why this is the case, compare the uninformative case ( = 1=2; r= 1:633andg = 0:775) to the polar opposite ( _!1=2.) Rel-ative to the former, in the latter i) low-equity intermediaries (i.e., those for whome <1=2) are perfectly identi…ed as excess risk takers but are charged accordingly (i.e., r = 2, g = 1, so that = 2 (1 1=2) = 1:), and ii), high-equity intermediaries enjoy lower face interest rates, which induces some of them to behave imprudently (while they would be prudent when charged the high face rate that prevails in the uninformative case.)

1,5 1,6 1,7 1,8 1,9 2 0 ,5 ₀,6 ₀,7 ₀,8 ₀,9 0, 999 pi r r_b r_g 0,5 0,6 0,7 0,8 0,9 1 0 ,5 0 ,6 0 ,7 0 ,8 0 ,9 0, 999 g_b g_g g_average pi

Figure 4. Face interest rates and shares of imprudent intermediaries in market =b; g; as a function of the precision of the signal, :

6

Impact of capital requirements

In this section, we explore the e¤ect on systemic risk of imposing capital constraints on intermediaries’behaviour. For tractability, we carry out our analysis under the same para-metric speci…cation as in the previous section (i.e., f(:)is uniformly distributed over [0;1]; p = w = 1=2; = 1, Rh₁ = 4, and the supply of loanable funds is inelastic at w = 1=2.) We consider two simple forms of capital ratios: a ‘naïve’capital ratio based exclusively on balance-sheet size, and a risk-based capital ratio that ties the stringency of the ratio to the level of portfolio diversi…cation achieved by the intermediary. As we show, the former may turn out to raise than lower aggregate risk-taking, due to the impact of constrained …rms on the equilibrium interest rate. However, in our parametric example risk-based capital ratio are e¤ective at curbing systemic risk (assuming that they are feasible.)

Asset size-based capital ratios. We …rst consider the impact of a simple (naïve) capital ratio prescribing that intermediaries must hold as initial equity at least some pre-speci…ed fraction ₂(0;1) of total assets. That is, we impose

e Pxi: (39)

The constraint (39) might be binding or not, depending on the equity level of each individual intermediary. In order to keep the analysis concise, we focus on the (realistic) case where is (i) su¢ ciently high for (39) to be binding at least for some intermediaries, given the equity distribution f(e); and (ii) su¢ ciently low for (39) not to be binding for intermediaries that would spontaneously choose the prudent portfolio in the absence of a capital constraint (essentially because those are su¢ ciently capitalised in the …rst place.) In short, we focus on the case where (39) may limit the leverage and investment of some (but not necessarily all) of the intermediaries that would behave imprudently in the absence of the constraint. Under our parametric equity distribution, this amounts to assuming that is positive but small.

We …rst show that the constraint may be e¤ective at limiting the leverage of imprudent intermediaries –i.e., when (39) is binding–, but not at inducing portfolio diversi…cation by those intermediaries. Second, we show that by limiting the leverage of low-equity interme-diaries, the capital ratio exerts a downward pressure on the equilibrium face interest rate, which induces some of the originally prudent intermediaries to become imprudent. In

con-sequence, a capital ratio purely based on size may ultimately raise, rather than lowers, the share of imprudent intermediaries in the economy.

Impact of the ratio on (low-equity) intermediaries. From (39), the leverage of an intermediary facing a binding capital constraint is

^_{b(e) =} P_x

i e=e 1 1 >0: (40)

For low values of , (39) is potentially binding for low-equity intermediaries, i.e. those who heavily resort on leverage in the absence of a capital constraint. As discussed above, we focus on the case where is su¢ ciently low for the constraint to be potentially binding only for imprudent intermediaries. Under our parameters, in the absence of capital constraint those imprudent intermediaries would choose a leverage of (see (19)):

b (r; e) = 2 1

2r e: (41)

The capital ratio is binding if and only if^b(e)< b (r; e), that is, if and only if

e < 2 1

2r ^e(r; ):

On the liability side, an intermediary facing a binding capital constraint chooses a lower level of leverage than it would otherwise. On the asset side, does the constraint alter its portfolio choice? The answer is no. To see this, compare the values of a prudent and an imprudent intermediary with total assets given by Pxi = e= (i.e., the intermediary is

constrained.) From our analysis in Section 3 and under our parameters speci…cation, the former and the latter are given by, respectively

~

V (e) =re+ e (2 r) c e ; V~ (e) = 1 2 re+

e

(4 r) c e : (42) Since V~ (e) > V~ (e), an intermediary facing a binding constraint always chooses the imprudent portfolio (~x1;x~2) = (e= ;0):

Loanable funds equilibrium. Under our maintained assumption that is small, we have ^

e(r) < e~(r), so that the capital constraint may only be binding for originally imprudent intermediaries. Then, the demand for loanable funds by the intermediary sector is given by:

Bd(r; ) = Z e^(r; ) 0 ^_{b(e)dF (e) +}Z e~(r) ^ e(r; ) b (r; e)dF (e) + Z 1 ~ e(r) b (r; e)dF (e):

Using (40)–(41), the fact that dF (e) = de and rearranging, we may rewrite the latter expression as: Bd(r; ) = 3 2 3 8r 2 2 2 1 2r 2 : (43)

The face interest rate that clears the market equatesBd_{(r; )with the aggregate supply}

of funds, w = 1=2. When = 0 (our baseline scenario), we again have r = p8=3 (the solution to3=2 3r2=8 = 1=2)and g(r; ) = 2 3r=4 = 0:775 (see equation (34) above.) As rises and constrains the leverage choices of more and more intermediaries, the aggregate demand curve Bd_{(r; ) shifts down. Given the vertical loan supply curve B}s _{= 1=2, the}

equilibrium face interest rate r must go down. Solving the equation Bd(r; ) = 1=2 for r, we indeed obtain the decreasing interest rate function

r( ) = 4 3 + + r 3 5 2 ! : (44)

Finally, since intermediaries facing a binding capital constraint choose the imprudent portfolio (~x1;x~2) = (e= ;0), the share of imprudent intermediaries in the economy is given

by:

Z ~e(r) 0

de = ~e(r( )) = 2 3 4r( );

which is increasing in . To summarise, simple capital ratios based on balance-sheet size are (in our example) ine¤ective at limiting systemic risk. Quite on the contrary, by lowering the equilibrium face interest rate, the capital constraint worsens the risk-taking channel and induces imprudent behaviour by those intermediaries that would otherwise behave prudently.

Risk-based capital ratios One key reason for the ine¤ectiveness of simple capital ratios is that even though the ratio does limit some of the intermediaries’borrowing, it does not curb their risk-taking incentives on the asset side. Suppose now that the regulator (but not an outside lender) is able to observe the riskiness of intermediaries’portfolio and to set the capital ratio accordingly. For example, assume that the capital ratio e=Pxi is 2 (0;1)

for a prudent intermediary, but ~ > for an imprudent intermediary. Incorporating this risk-based capital ratios into the values of being prudent or imprudent in (42), we …nd that a constrained intermediary prefers to be prudent if and only if

re+ e pRh₁ r c e > p re+ e ~ R h 1 r c e ~ :

A su¢ ciently large value of ~ (relative to ) acts as a deterrent and induces prudent behaviour by constrained intermediaries, so that rather than ~ e¤ectively applies. Under this regulatory arrangement, the aggregate demand for loanable funds is as in (43), and consequently the equilibrium face interest rate as in (44). However, since intermediaries facing a binding constraint now behave prudently, the share of imprudent intermediaries in the economy is now

Z e~(r) ^ e(r; ) de= ~e(r) e^(r; ) = 2 3r( ) 4 2 1 2r( ) ;

where r( ) is given by (44). The latter expression is decreasing in , implying that a risk-based capital requirement is e¤ective at reducing economywide risk-taking (again, within our parametric example.)

7

Concluding remarks

In this paper, we have analysed the portfolio and leverage choices of limited-liability inter-mediaries and their implications for the level of aggregate risk and the way it responds to changes in economic conditions. The novelty of our framework relative to earlier analysis of intermediaries’risk-shifting behaviour is twofold. First, we emphasise the disciplining role of shareholders’ inside equity stake and the heterogeneities that it implies for their equi-librium balance sheets –both on the asset and liability sides. Second, and relatedly, we explicitly model changes in economywide risk-shifting along the “extensive margin” –i.e., due to changes in the number of intermediaries endogenously choosing to expose themselves to the risk of default–, in addition to the usual intensive margin –i.e., related to changes in their individual balance-sheet choices.

A important property of the model’s equilibrium is that it jointly determines the (com-mon) borrowing rate faced by intermediaries and the level of aggregate risk in the economy, due to the endogenous sorting of intermediaries along the equity dimension. Unsurprisingly, intermediaries with low shareholders’stake are more likely to behave imprudently than those with high inside equity stake. More interestingly, the sorting of intermediaries his itself af-fected by the interest rate, with falling interest associated with a rising number of imprudent intermediaries and aggregate risk. For this reason, exogenous factors that a¤ect the market for loanable funds (e.g., international capital ‡ows) have a direct impact on the level of risk

generated by the …nancial sector. Similarly, exogenous changes in the distribution of inter-mediaries’ capital a¤ect the equilibrium interest rate, aggregate risk, and the return that ultimate lenders can expect from ent